Currently, the Mozilla Firefox Beta channel and the Developer Edition channel both offer the same version of Firefox.
What is the difference between the Beta version and the Developer Edition?
Related, but different: What is the difference between Mozilla Firefox and Mozilla Firefox Developer Edition? (this other question involves the release channel, not the Beta channel)
Firefox used to have 4 release channels, Nightly>Aurora>Beta>Release. A couple of years ago, they rebranded the Aurora channel as "Developer Edition", to appeal to developers and to get more people using it.
Recently, the Aurora channel was phased out, and Developer Edition is more or less just Beta, but set up with some different defaults, like theme and available developer tools. DevEdition automatically sends feedback to Mozilla and allows to load unsigned add-ons.
Both versions have the same code base but different configs. Everything in Beta is meant to be shipped, but not in DevEdition. Beta is basically the next release of Firefox, about 6 weeks in advance.
DevEdition got the new debugger by default, forced multi process even if you had incompatible addons, had the new compact themes before they landed in beta. It is just Beta with more experimental features and tools enabled (see the list).
